Paper · May 19, 2026

Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

Mattia Fogagnolo, Giorgio Gatti, Alessandra Pluda

math.DGWider flowsarXiv:2605.20114v2
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Abstract

We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem for continuous Riemannian metrics with nonnegative scalar curvature in such IMCF sense.

Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsScalar curvatureStability & uniquenessPhysics

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper The asymptotic behaviour of p-capacitary potentials in Asymptotically Conical manifolds (Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri, 2022)5Benatti 2022 Hawking Mass Monotonicity for Initial Data Sets (Sven Hirsch, 2022)4Hirsch 2022 On the Isoperimetric Riemannian Penrose Inequality (Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri, 2022)2Benatti 2022 A topological gap theorem for the π₂-systole of positive scalar curvature 3-manifolds (Kai Xu, 2023)6Xu 2023 Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature (Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli et al., 2024)1Antonelli 2024 Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas (Luca Benatti, Alessandra Pluda and Marco Pozzetta, 2024)3Benatti 2024 3-Manifolds with positive scalar curvature and bounded geometry (Otis Chodosh, Yi Lai and Kai Xu, 2025)7Chodosh 2025 The σ-inverse mean curvature flow and the generalized Penrose conjecture (Conghan Dong, 2026)8Dong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

    Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli et al.Wider flows PDF

    Topics Inverse mean curvature flow, Physics, Scalar curvature · Same author Mattia Fogagnolo · Shared terms “continuous metrics”, “nonnegative scalar”, “manifold endowed”

  2. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows PDF

    Topics Inverse mean curvature flow, Physics, Scalar curvature · Same author Mattia Fogagnolo · Shared terms “Hawking mass”, “nonnegative scalar”

  3. Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

    Luca Benatti, Alessandra Pluda and Marco PozzettaWider flows PDF

    Topics Inverse mean curvature flow, Physics, Scalar curvature · Same author Alessandra Pluda · Shared terms “Hawking mass”, “nonnegative scalar”, “Riemannian manifolds”

  4. Hawking Mass Monotonicity for Initial Data Sets

    Sven HirschWider flows PDF

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “Hawking mass”

  5. The asymptotic behaviour of -capacitary potentials in Asymptotically Conical manifolds

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Same author Mattia Fogagnolo · Shared terms “Riemannian manifolds”

  6. A topological gap theorem for the -systole of positive scalar curvature 3-manifolds

    Kai XuWider flows PDF

    Topics Inverse mean curvature flow, 3-manifolds, Scalar curvature

  7. 3-Manifolds with positive scalar curvature and bounded geometry

    Otis Chodosh, Yi Lai and Kai XuWider flows PDF

    Topics Inverse mean curvature flow, 3-manifolds, Scalar curvature

  8. The -inverse mean curvature flow and the generalized Penrose conjecture

    Conghan DongWider flows PDF

    Topics Inverse mean curvature flow, Physics, Other curvature flows

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.